Let's say you went to some jungle or mountain in a different country, even if your provider wanted to offer global service, you just can't get a base station everywhere. Ultimately, wouldn't satellite technology be the best for global, mobile phone service?

Satellites are unreliable as they get affected by cloud cover and signal interference from other satellites and they cost a heck of a lot more than base stations hence why there isn't a global cellphone network using them 100%

Going global with our current technology is the best idea, yes. As was mentioned, there are the satellite problems, but also there are far more legal issues with building Satellites and sending them into space. I think it's just for simplicity's sake.